Cumulative product of (1-previous_record)*current_record

A straightforward math approach using cumprod should work

> c(1, head(cumprod(1 - rate), -1)) * rate
 [1] 0.01000000 0.01970100 0.02881885 0.03709807 0.04432372 0.05033049
 [7] 0.05500858 0.05830607 0.06022773 0.06083074

If you want to practice with recursions, you can try the method below

f <- function(v, k = length(v)) {
    if (k == 1) {
        return(v[k])
    }
    u <- f(v, k - 1)
    c(u, tail(u, 1) * (1 / v[k - 1] - 1) * v[k])
}

such that

> f(rate)
 [1] 0.01000000 0.01970100 0.02881885 0.03709807 0.04432372 0.05033049
 [7] 0.05500858 0.05830607 0.06022773 0.06083074

In case you are still interested in how to do it with purrr::reduce family of functions. Here are two solutions:

• In every iteration if you multiply the accumulated/previous value by (1/previous value of rate - 1) * (current value of rate) you will get the desired output in every row

library(purrr)

accumulate2(dat$rate[-nrow(dat)], dat$rate[-1], .init = dat$rate[1], 
           ~ ..1 * (1/..2 - 1) * ..3) %>%
  simplify()

 [1] 0.01000000 0.01970100 0.02881885 0.03709807 0.04432372 0.05033049 0.05500858 0.05830607
 [9] 0.06022773 0.06083074

And also in base R we could do the following:

Reduce(function(x, y) {
  x * (1/dat$rate[y - 1] - 1) * dat$rate[y]
}, init = dat$rate[1], 
seq_len(nrow(dat))[-1], accumulate = TRUE)

 [1] 0.01000000 0.01970100 0.02881885 0.03709807 0.04432372 0.05033049 0.05500858 0.05830607
 [9] 0.06022773 0.06083074

cumprod to the rescue (hat-tip to @Cole for simplifying the code):

dat$rate * c(1, cumprod(1 - head(dat$rate, -1)))

The logic is that you are essentially doing a cumulative product of 1 - dat$rate, multiplied by the current step.
At the first step, you can just keep the existing value, but then you need to offset the two vectors so that the multiplication gives the desired result.

Proof:

out <- c(
dat$rate[1],
(1-dat$rate[1])*dat$rate[2],
(1-dat$rate[1])*(1-dat$rate[2])*dat$rate[3],
(1-dat$rate[1])*(1-dat$rate[2])*(1-dat$rate[3])*dat$rate[4],
(1-dat$rate[1])*(1-dat$rate[2])*(1-dat$rate[3])*(1-dat$rate[4])*dat$rate[5],
(1-dat$rate[1])*(1-dat$rate[2])*(1-dat$rate[3])*(1-dat$rate[4])*(1-dat$rate[5])*dat$rate[6],
(1-dat$rate[1])*(1-dat$rate[2])*(1-dat$rate[3])*(1-dat$rate[4])*(1-dat$rate[5])*(1-dat$rate[6])*dat$rate[7],
(1-dat$rate[1])*(1-dat$rate[2])*(1-dat$rate[3])*(1-dat$rate[4])*(1-dat$rate[5])*(1-dat$rate[6])*(1-dat$rate[7])*dat$rate[8],
(1-dat$rate[1])*(1-dat$rate[2])*(1-dat$rate[3])*(1-dat$rate[4])*(1-dat$rate[5])*(1-dat$rate[6])*(1-dat$rate[7])*(1-dat$rate[8])*dat$rate[9],
(1-dat$rate[1])*(1-dat$rate[2])*(1-dat$rate[3])*(1-dat$rate[4])*(1-dat$rate[5])*(1-dat$rate[6])*(1-dat$rate[7])*(1-dat$rate[8])*(1-dat$rate[9])*dat$rate[10]
)

all.equal(
  dat$rate * c(1, cumprod(1 - head(dat$rate, -1))),
  out
)
#[1] TRUE